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Metamath Proof Explorer

Theorem 3netr4d

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012) (Proof shortened by Wolf Lammen, 21-Nov-2019)

	Ref	Expression
Hypotheses	3netr4d.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
	3netr4d.2	⊢ ( 𝜑 → 𝐶 = 𝐴 )
	3netr4d.3	⊢ ( 𝜑 → 𝐷 = 𝐵 )
Assertion	3netr4d	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3netr4d.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
2		3netr4d.2	⊢ ( 𝜑 → 𝐶 = 𝐴 )
3		3netr4d.3	⊢ ( 𝜑 → 𝐷 = 𝐵 )
4	2 1	eqnetrd	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
5	4 3	neeqtrrd	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )